Property of a set of sentences from which a contradiction can be derived under the given consequence relation, or equivalently of a theory that proves both a formula and its negation.

Principle

A theory is inconsistent if the consequence relation yields both φ and ¬φ for some φ; under classical logic this typically entails triviality (explosion), though alternative logics may block explosion.

Demonstration

In a deductive system, if from axioms A one can derive both p and ¬p then A is inconsistent; for example, adding the axioms {p, ¬p} to any theory makes it inconsistent and enables derivation of arbitrary sentences in classical systems.

Misapplication

Equating semantic unsatisfiability with inconsistency without reference to the consequence relation: a set may be semantically unsatisfiable yet not exhibit a syntactic contradiction in an incomplete proof system.

Consequence

Inconsistency compromises reliable inference: in classical logics it leads to explosion where any sentence becomes derivable, forcing revision of axioms or adoption of paraconsistent strategies to continue reasoning.

Reversal

Consistency is the reversal: a consistent set does not yield contradictions; examining minimal inconsistent subsets (unsatisfiable cores) reverses the diagnosis to locate problematic axioms.

Boundary

Inconsistency is defined relative to the chosen consequence relation and logic; paraconsistent logics, relevance logics, or restricted proof systems modify whether a derived contradiction collapses the theory.

Semantic Tension

Inconsistency vs unsatisfiability: inconsistency is syntactic derivability of contradiction, whereas unsatisfiability is semantic absence of models; they align in sound and complete frameworks but differ in practice.

Synthesis

Inconsistency marks a theory from which contradictions follow under its consequence relation; it signals a need for repair, restricts trustworthy inference, and interacts with semantic unsatisfiability and choices of logic.